Tangent Lines and Rates of Change



Unit 4: Rules of Differentiation

|DAY |TOPIC |ASSIGNMENT |

| | | |

|1 |Power Rule |p. 23 |

| | | |

|2 |Power Rule Again |p. 24 |

| | | |

|3 |Even More Power Rule |p. 25 |

| | | |

|4 |QUIZ 1 | |

| | | |

|5 |Rates of Change |p. 26-27 |

| | | |

|6 |Rates of Change |p. 28-29 |

| | | |

|7 |Quiz 2 | |

| | | |

|8 |Product Rule |p. 30-31 |

| | | |

|9 |Quotient Rule |p. 32-33 |

| | | |

|10 |Both Product and Quotient Rules |p. 34 |

| | | |

|11 |Quiz 3 | |

| | | |

|12 |Chain Rule |p. 35 |

| | | |

|13 |More Chain Rule |p. 36 |

| | | |

|14 |Still More Links |p. 37-40 |

| | | |

|15 |Quiz 4 | |

| | | |

|16 |Higher Order Derivatives |p. 41-42 |

| | | |

|17 |Even Higher Derivatives |p. 43-44 |

| | | |

|18 |Review |Worksheet (Passed out in class) |

| | | |

|19 |Test | |

3.1 Techniques of Differentiation

Learning Objectives

A student will be able to:

• Use various techniques of differentiations to find the derivatives of various functions.

• Compute derivatives of higher orders.

Up to now, we have been calculating derivatives by using the definition. In this section, we will develop formulas and theorems that will calculate derivatives in more efficient and quick ways. It is highly recommended that you become very familiar with all of these techniques.

The Derivative of a Constant

If [pic]where [pic]is a constant, then [pic].

In other words, the derivative or slope of any constant function is zero.

Proof:

[pic]

Example 1:

If [pic]for all [pic], then [pic]for all [pic]. We can also write [pic].

The Power Rule

If [pic]is a positive integer, then for all real values of [pic]

[pic]The proof of the power rule is omitted in this text, but it is available at and also in video form at Khan Academy Proof of the Power Rule. Note that this proof depends on using the binomial theorem from Precalculus.

[pic].

Example 2:

If [pic], then

[pic]

and

[pic]

The Power Rule and a Constant

If [pic]is a constant and [pic]is differentiable at all [pic], then

[pic]

In simpler notation,

[pic]

In other words, the derivative of a constant times a function is equal to the constant times the derivative of the function.

Example 3:

[pic]

Example 4:

[pic]

Derivatives of Sums and Differences

If [pic]and [pic]are two differentiable functions at [pic], then

[pic]

and

[pic]

In simpler notation,

[pic]

[pic]

The Product Rule

If [pic]and [pic]are differentiable at [pic], then

[pic]

In a simpler notation,

[pic]

The derivative of the product of two functions is equal to the first times the derivative of the second plus the second times the derivative of the first.

Keep in mind that

[pic]

Example 7:

Find [pic]for [pic]

Solution:

There are two methods to solve this problem. One is to multiply the product and then use the derivative of the sum rule. The second is to directly use the product rule. Either rule will produce the same answer. We begin with the sum rule.

[pic]

Taking the derivative of the sum yields

[pic]

Now we use the product rule,

[pic]

which is the same answer.

The Quotient Rule

If [pic]and [pic]are differentiable functions at [pic]and [pic], then

[pic]

In simpler notation,

[pic]

The derivative of a quotient of two functions is the bottom times the derivative of the top minus the top times the derivative of the bottom all over the bottom squared.

Keep in mind that the order of operations is important (because of the minus sign in the numerator) and

[pic]

Example 8:

Find [pic]for

[pic]

Solution:

[pic]

Example 9:

At which point(s) does the graph of [pic]have a horizontal tangent line?

Solution:

Since the slope of a horizontal line is zero, and since the derivative of a function signifies the slope of the tangent line, then taking the derivative and equating it to zero will enable us to find the points at which the slope of the tangent line equals to zero, i.e., the locations of the horizontal tangents.

[pic]

Multiplying by the denominator and solving for [pic],

[pic]

Therefore the tangent line is horizontal at [pic]

Higher Derivatives

If the derivative [pic]of the function [pic]is differentiable, then the derivative of [pic], denoted by [pic], is called the second derivative of [pic]. We can continue the process of differentiating derivatives and obtain third, fourth, fifth and higher derivatives of [pic]. They are denoted by [pic], [pic], [pic], [pic], [pic]

Example 10:

Find the fifth derivative of [pic].

Solution:

[pic]

Example 11:

Show that [pic]satisfies the differential equation [pic]

Solution:

We need to obtain the first, second, and third derivatives and substitute them into the differential equation.

[pic]

Substituting,

[pic]

which satisfies the equation.

Review Questions

Use the results of this section to find the derivatives [pic].

1. [pic]

2. y = [pic]

3. [pic]

4. [pic] (where a and b are constants)

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. Newton’s Law of Universal Gravitation states that the gravitational force between two masses (say, the earth and the moon), m and M, is equal to their product divided by the square of the distance r between them. Mathematically, [pic] where G is the Universal Gravitational Constant [pic]. If the distance r between the two masses is changing, find a formula for the instantaneous rate of change of F with respect to the separation distance r.

12. Find [pic], where [pic] is a constant.

13. Find[pic], where [pic].

Review Answers

(some answers simplify further than the given responses)

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

13. -120

Constant and Power Rule Practice

Use the Constant and Power Rules to find the derivative.

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. [pic] 8. [pic]

9. [pic] 10. [pic]

11. [pic] 12. [pic]

13. [pic] 14. [pic]

15. [pic] 16. [pic]

Find the value of the derivative of the function at the indicated point.

17. [pic] 18. [pic]

Find the equation of the tangent line to the graph of the function at the indicated point

19. [pic] 20. [pic]

Answers:

1. [pic] 2. [pic] 3. [pic] 4. [pic] 5. [pic]

6. [pic] 7. [pic] 8. [pic] 9. [pic] 10. [pic]

11. [pic] 12. [pic] 13. [pic] 14. [pic]

15. [pic] 16. [pic] 17. [pic] 18. [pic]

19. [pic] 20. [pic]

Power Rule Practice

Find the derivative of each function. In your answers, rational exponents are OK, negative exponents are not.

1.) [pic]

2.) [pic]

3.) [pic]

4.) [pic] POWER RULE: [pic]

5.) [pic]

6.) [pic]

7.) [pic]

Answers:

1.) [pic]

2.) [pic]

3.) [pic]

4.) [pic]

5.) [pic]

6.) [pic]

7.) [pic]

Mo’ Power Rule Practice

Find the derivative of each function. Make sure your answers are factored completely. If a point is given, find the value of the derivative at that point.

1.) [pic] Point: [pic]

2.) [pic] Point: (0, 1)

3.) [pic] Point: (1, –3)

4.) [pic]

5.) [pic] Point: (7, 350)

6.) [pic]

7.) [pic] Point: (–2, –512)

8.) [pic]

Answers:

1.) [pic]

2.) [pic]

3.) [pic]

4.) [pic]

5.) [pic]

6.) [pic]

7.) [pic]

8.) [pic]

Product Rule Practice

Find the derivative using the Product Rule. Final answer should be in simplest form.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

Determine the point(s), if any, at which the graph of the function has a horizontal tangent line.

5. [pic] 6. [pic]

Find the derivative. Do not use the Product Rule.

7. [pic] 8. [pic]

Answers:

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. none 7. [pic]

8. [pic]

Product Rule Practice

Find the derivative of each function. Make sure your answers are factored completely. If a point is given, find the value of the derivative at that point.

1.) [pic] Point: (1, 2)

2.) [pic] Point: [pic]

3.) [pic] Point: (4, 6)

4.) [pic]

5.) [pic] PRODUCT RULE: [pic]

6.) [pic] Point: (2, 36)

7.) [pic]

8.) [pic]

Answers:

1.) [pic]

2.) [pic]

3.) [pic]

4.) [pic]

5.) [pic]

6.) [pic]

7.) [pic]

8.) [pic]

Quotient Rule Practice

Use the Quotient Rule to find the derivative. Final answers should be in simplest form.

1. [pic]

2. [pic]

3. [pic]

Find the equation of the line tangent to [pic] at the indicated point.

4. [pic] Point: (6, 6)

5. [pic] Point: [pic]

Find the derivative without the use of the Product or Quotient Rules. Give simplified final answers.

6. [pic]

7. [pic]

Answers:

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic] 6. [pic]

7. [pic]

Quotient Rule Practice

Find the derivative of each function. Make sure your answers are factored completely. If a point is given, find the value of the derivative at that point.

1.) [pic] Point: (6, 6)

2.) [pic] Point: [pic]

3.) [pic] QUOTIENT RULE: [pic]

4.) [pic]

5.) [pic] (Do not use the product or quotient rules.)

6.) [pic] (Do not use the quotient rule.)

7.) [pic] Point: (2, 1)

8.) [pic]

Answers:

|1.) [pic] |5.) [pic] |

| | |

|2.) [pic] |6.) [pic] |

| | |

|3.) [pic] |7.) [pic] |

| | |

|4.) [pic] |8.) [pic] |

Practice Problems (Constant, Power, Product & Quotient Rules)

Differentiate. Remember to simplify the function to make differentiating easier. Final answers should be in simplest form.

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. [pic] 8. [pic]

9. [pic] 10. [pic]

11. [pic] 12. [pic]

Find the equation of the line tangent to the function at the indicated [pic] value.

13. [pic] [pic]

Find the slope of the graph at the indicated point.

14. [pic] [pic]

Find the point(s), if any, at which the graph of the function has a horizontal tangent.

15. [pic]

Answers:

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. [pic] 7. [pic] 8. [pic]

9. [pic] 10. [pic] 11. [pic]

12. [pic] 13. [pic] 14. [pic]

15. [pic]

3.3 The Chain Rule

Learning Objectives

A student will be able to:

• Know the chain rule and its proof.

• Apply the chain rule to the calculation of the derivative of a variety of composite functions.

We want to derive a rule for the derivative of a composite function of the form [pic]in terms of the derivatives of f and g. This rule allows us to differentiate complicated functions in terms of known derivatives of simpler functions.

The Chain Rule

If [pic]is a differentiable function at [pic]and [pic]is differentiable at [pic], then the composition function [pic]is

differentiable at [pic]. The derivative of the composite function is:

[pic]

Another way of expressing, if [pic]and [pic], then

[pic]

And a final way of expressing the chain rule is the easiest form to remember: If [pic]is a function of [pic]and [pic]is a function of [pic], then

[pic]

Example 1:

Differentiate [pic]

Solution:

Using the chain rule, let [pic]Then

[pic]

The example above is one of the most common types of composite functions. It is a power function of the type

[pic]

The rule for differentiating such functions is called the General Power Rule. It is a special case of the Chain Rule.

The General Power Rule

if

[pic]

then

[pic]

In simpler form, if

[pic]

then

[pic]

Example 2:

What is the slope of the tangent line to the function [pic]that passes through point [pic]?

Solution:

We can write [pic]This example illustrates the point that [pic]can be any real number including fractions. Using the General Power Rule,

[pic]

To find the slope of the tangent line, we simply substitute [pic]into the derivative:

[pic]

Example 3:

Find [pic]for [pic].

Solution:

The function can be written as [pic]Thus

[pic]

Example 4:

Find [pic]for [pic]

Solution:

Let [pic]By the chain rule,

[pic]

where [pic]Thus

[pic]

Example 5:

Find [pic]for [pic]

Solution:

This example applies the chain rule twice because there are several functions embedded within each other.

Let [pic]be the inner function and [pic]be the innermost function.

[pic]

Using the chain rule,

[pic]

Notice that we used the General Power Rule and, in the last step, we took the derivative of the argument.

Multimedia Links

For an introduction to the Chain Rule (5.0), see Khan Academy, Calculus: Derivatives 4: The Chain Rule (9:11)[pic].

For more examples of the Chain Rule (5.0), see [ Math Video Tutorials by James Sousa The Chain Rule: Part 1 of 2 (8:45)[pic]. and [ Math Video Tutorials by James Sousa The Chain Rule: Part 2 of 2 (8:35)[pic].

Review Questions

Find [pic].

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

Review Answers

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic] or [pic]

9. [pic]

10. [pic]

Or [pic]

11. [pic]

Chain Rule Practice

Differentiate. Simplify answers.

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. [pic] 8. [pic]

9. [pic] 10. [pic]

11. Find the equation of the tangent line to the graph of [pic] at the point [pic].

12. Find the slope of [pic] when [pic].

Answers:

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6.[pic]

7. [pic] 8. [pic]

9. [pic] 10. [pic]

11.

[pic] 12. [pic]

Day 1

Find the derivative of each function.

1) [pic] 2) [pic]

3) [pic] 4) [pic]

5) [pic] 6) [pic]

7) [pic] 8) [pic]

9) [pic]

10) [pic]

11) [pic] 12) [pic]

13) [pic] 14) [pic]

15) [pic] 16) [pic]

17) [pic] 18) [pic]

Day 2

Find f’(x).

1) [pic] 2) [pic]

3) [pic] 4) [pic]

5) [pic] 6) [pic]

7) [pic] 8) [pic]

9) [pic] 10) [pic]

Find the value of the derivative at the indicated point.

11) [pic] 12) [pic]

13) [pic] 14) [pic]

Day 3

Determine the point(s), if any, at which the graph of the function has a horizontal tangent line.

1) [pic] 2) [pic]

3) [pic] 4) [pic]

5) The variable cost for manufacturing an electrical component is $7.75 per unit, and the fixed cost is $500. Write the cost C as a function of x, the number of units produced. Show that the derivative of this cost function is constant and is equal to the variable cost. (This is called the marginal cost. A marginal cost is the derivative of the cost)

6) A college club raises funds by selling candy bars for $1.00 each. The club pays $0.60 for each candy bar and has annual fixed costs of $250. Write the profit P as a function of x, the number of candy bars sold. Show that the derivative of the profit function is a constant and that it is equal to the profit on each candy bar sold. (This is called the marginal profit. A marginal profit is the derivative of the profit)

Using what you have learned about derivatives as well as your calculator. Find any points on the given interval of x for which the function f has a horizontal tangent line.

7) [pic] 8) [pic]

Day 5

1) The effectiveness E (on a scale from 0 to 1) of a pain-killing drug t hours after entering the bloodstream is given by [pic] Find the average rate of change E on the intervals [2, 3] and compare this with the instantaneous rates of change at the endpoints of the interval.

2) At 0° Celsius, the heat loss H (in kilocalories per square meter per hour) from a person’s body can be modeled by [pic], where v is the wind speed (in meters per second). Find the instantaneous rate of change H when v =2 and when v = 5.

3) The height s (in feet) at a time t (in seconds) of a silver dollar dropped from the top of the Washington Monument is [pic].

a) Find the average velocity on the interval [2,3].

b) Find the instantaneous velocity when t = 2 and when t = 3.

c) How long will it take the dollar to hit the ground?

d) Find the velocity of the dollar when it hits the ground.

4) The height s (in feet) of an object fired straight up from the ground level with an initial velocity of 200 feet per second is given by [pic], where t is the time (in seconds).

a) How fast is the object moving after one second?

b) During which interval of time is the speed decreasing?

c) During which interval of time is the speed increasing?

5) The position s (in feet) of an accelerating car is [pic], where t is the time (in seconds). Find the velocity of the car at the following times.

a) t = 0 b) t = 1

c) t = 4 d) t = 9

6) Given the position of an object is given by [pic] find the equation for the velocity of the object.

Day 6

1) Given the cost function [pic], find the marginal cost function.

2) Given the revenue function [pic], find the marginal revenue function.

3) Given the profit function [pic], find the marginal profit function.

4) The revenue (in dollars) from producing x units of a product is [pic]

a) Find the additional revenue when production is increased from 15,000 units to 15,001 units.

b) Find the marginal revenue when x = 15,000.

c) Compare your results from parts a & b.

5) The cost (in dollars) of producing x units of a product is [pic]

a) Find the additional cost when the production increases from nine to ten units.

b) Find the marginal cost when x = 9.

c) Compare your results from parts a & b.

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6) The profit (in dollars) from selling x units of a product is [pic]

a) Find the additional profit when the sales increase from seven to eight units.

b) Find the marginal revenue when x = 7.

c) Compare your results from parts a & b.

7) The profit (in dollars) for selling x units of a product is given by [pic]

Find the marginal profit for the following sales.

a) x = 10 b) x = 20 c) x =23 d) x = 25

Day 8

Differentiate each function.

1) [pic] 2) [pic]

3) [pic] 4) [pic]

5) [pic] 6) [pic]

Find the value of the derivative of the function at the indicated point.

7) [pic] 8) [pic]

9) [pic]

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10) Find the equation of the line tangent to [pic]at (0,2)

11) Find the equation of the line tangent to [pic] at (1, -3)

Day 9

Differentiate each function.

1) [pic] 2) [pic]

3) [pic] 4) [pic]*Hint: multiply first then use Quo.Rule

5) [pic] 6) [pic]

5-You don’t have to use Quo.Rule if you don’t

want. You can distribute instead if you like.

Or you can use Quo.Rule too if you like.

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Find the value of the derivative of the function at the indicated point.

7) [pic] 8) [pic]

9) [pic]

10) Find the equation of the line tangent to [pic]at (2,1)

11) Find the equation of the line tangent to [pic] at (2,1/3)

Day 10

Find the derivative of each function.

1) [pic] 2) [pic]

2) [pic] 4) [pic]

Hint: You will need to use both! Or you will

Have to multiply carefully then use the Quo.Rule.

Find the point(s), if any, at which the graph of f has a horizontal tangent.

5) [pic] 6) [pic]

7) The percent P of defective parts produced by a new employee t days after the employee starts work can be modeled by [pic]. Find the rate of change P when t = 10.

Day 12

Find the derivative of each function.

1) [pic] 2) [pic]

3) [pic] 4) [pic]

5) [pic] 6) [pic]

7) [pic] 8) [pic]

9) [pic] 10) [pic]

11) Find the equation of the line tangent to [pic]at x = 2.

Find the derivative of each function. (Hint: Rewrite them so they are in terms of negative powers instead of fractions.

12) [pic] 13) [pic]

Day 13

Find the derivative of each function.

1) [pic] 2) [pic]

3) [pic] 4) [pic]

5) [pic] 6) [pic]

7) [pic] 8) [pic]

9) [pic] 10) [pic]

11) Find the equation of the line tangent to [pic]at x = 2. Hint: Use product rule & chain rule

Find the derivative of each function. (Hint: Rewrite them so they are in terms of negative powers instead of fractions.

12) [pic] 13) [pic]

Day 14

Differentiate each function. You may have to re-write, use product rule, quotient rule and/or chain rule.

1) [pic] 2) [pic]

3) [pic] 4) [pic]

5) [pic] 6) [pic]

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7) [pic] 8) [pic]

9) [pic] 10) [pic]

11) [pic] 12) [pic]

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13) [pic] 14) [pic]

15) [pic] 16) [pic]

17) [pic] 18) [pic]

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19) [pic]

20) [pic]

Day 16

Find the second derivative of each function.

1) [pic] 2) [pic]

3) [pic] 4) [pic]

5) [pic] 6) [pic]

7) [pic] 8) [pic]

Find the third derivative of each function

9) [pic] 10) [pic]

Find the indicated derivative of each function.

11) [pic] 12) [pic]

13) [pic]

Find the x-value(s) where the second derivative equals zero. f ’’ (x) = 0

14) [pic] 15) [pic]

Day 17

Find the second derivative of each function.

1) [pic] 2) [pic]

3) [pic] 4) [pic]

5) [pic] 6) [pic]

7) [pic] 8) [pic]

Find the third derivative of each function

9) [pic] 10) [pic]

Find the indicated derivative of each function.

11) [pic] 12) [pic]

13) [pic]

Find the x-value(s) where the second derivative equals zero. f ’’ (x) = 0

14) [pic] 15) [pic]

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